add memoization

dynamic_programming/README.md

@@ -2,7 +2,7 @@
 
 <br>
 
-* take a recursive algorithm and cache overlapping problems (repeated calls).
+* dynamic programming is the process of taking a recursive algorithm and cache overlapping problems (repeated calls).
 * the runtime is given by the number of calls.
 * **top-down**: how can we divide the problem into sub-problems?
     * top-down dynamic programming is called **memoization**.
@@ -10,3 +10,46 @@
 
 <br>
 
+---
+
+### recursion
+
+<br>
+
+* recursion is an approach to solving problems using a function that calls itself as a subroutine. every time the function calls itself, it reduces the problem into subproblems. the recursion calls continue until it reaches a point where the subproblem can be solved without further recursion.
+* a recursive function should have the following properties so it does not result in an infinite loop:
+   * one or more base cases (a terminating scenario that does not use recursion to produce an answer)
+   * a set of rules, also known as recurrence relation, that reduces all other cases towards the base case.
+* there can be multiple places where the function may call itself. 
+* any recursion problem can be solved iteratively and vice-versa.
+
+<br>
+
+#### vizualing the stack
+
+<br>
+
+* to visualize how the stack operates during recursion calls, check the example below where we reverse a string:
+
+```python
+def reverse(s):
+   if len(s) == 0:
+      return s
+   else:
+      return reverse(s[1:]) + s[0]
+```
+
+<br>
+
+---
+
+### memoization
+
+<br>
+
+* memoization is an optimization technique that avoids recursion's duplicate calculations.
+* it's primarily used to speed up code by storing the intermediate results in a cache so that it can be reused later.
+* for example, a hash table can be used as a cache and should be passed along each subroutine call.
+
+
+